Numerical optimization for the calculus of variations by gradients on non-Hilbert Sobolev spaces using conjugate gradients and normalized differential equations of steepest descent

The purpose of this paper is to illustrate the application of numerical optimizationmethods for nonquadratic functionals defined on non-Hilbert Sobolev spaces. These methods use a gradient defined on a norm-reflexive and hence strictly convex normed linear space. This gradient is defined by Michael Golomb and Richard A. Tapia in [M. Golomb, R.A. Tapia, Themetric gradient in normed linear spaces, Numer. Math. 20 (1972) 115–124]. It is also the same gradient described by Jean-Paul Penot in [J.P. Penot, On the convergence of descent algorithms, Comput. Optim. Appl. 23 (3) (2002) 279–284]. In this paper we shall restrict our attention to variational problems with zero boundary values. Nonzero boundary value problems can be converted to zero boundary value problems by an appropriate transformation of the dependent variables. The original functional changes by such a transformation. The connection to the calculus of variations is: The notion of a relative minimum for the Sobolev norm for p positive and large and with only first derivatives and function values is related to the classical weak relative minimum in the calculus of variations. The motivation for minimizing nonquadratic functionals on these non-Hilbert Sobolev spaces is twofold. First, a norm equivalent to this Sobolev norm approaches the norm used for weak relative minimums in the calculus of variations as p approaches infinity. Secondly, the Sobolev norm is both norm-reflexive and strictly convex so that the gradient for a non-Hilbert Sobolev space consists of a singleton set; hence, the gradient exists and is unique in this non-Hilbert normed linear space. Two gradient minimization methods are presented here. They are the conjugate gradient methods and an approach that uses differential equations of steepest descent. The Hilbert space conjugate gradient method of James Daniel in [J. Daniel, The Approximate Minimization of Functionals, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971], is one conjugate gradient method extended to a conjugate gradient procedure for a non-Hilbert normed linear space. As a reference see Ivie Stein Jr., [I. Stein Jr., Conjugate gradient methods in Banach spaces, Nonlinear Anal. 63 (2005) e2621–e2628] where local convergence theorems are given. The approach using a differential equation of steepest descent is motivated and described by James Eells Jr. in [J. Eells Jr., A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966) 751–807]. Also a normalized differential equation of steepest descent is used as a numericalminimization procedure in connectionwith starting methods such as higher order Runge–Kuttamethods described byBaylis Shanks in [E. Baylis Shanks, Solutions of differential equations by evaluations of functions, Math. Comput. 20 (1966) 21–38], and higher order multi-step methods such as Adams–Bashforth described ∗ Corresponding address: Department of Mathematics, Mail Stop 942, University of Toledo, 2801 W. Bancroft St., 43606-3390 Toledo, OH, USA. Tel.: +1 419 53